180624

Ruby


(1 - b*x)^(-1/A003557(b))

b が1728以下で、A003557(b) > 1 となる場合について
出力してみた。

require 'prime'

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end

def A003557(n)
  s = 1
  n.prime_division.each{|i| s *= i[0] ** (i[1] - 1)}
  s
end

def A(a, k, n)
  [1] + (1..n).map{|i| (k / a) ** i * (0..i - 1).inject(1){|s, j| s * (a * j + 1)} / f(i)}
end

n = 6
(1..1728).each{|i|
  a = A003557(i)
  p [i, a, A(a, i, n)] if a > 1
}

出力結果
[4, 2, [1, 2, 6, 20, 70, 252, 924]]
[8, 4, [1, 2, 10, 60, 390, 2652, 18564]]
[9, 3, [1, 3, 18, 126, 945, 7371, 58968]]
[12, 2, [1, 6, 54, 540, 5670, 61236, 673596]]
[16, 8, [1, 2, 18, 204, 2550, 33660, 460020]]
[18, 3, [1, 6, 72, 1008, 15120, 235872, 3773952]]
[20, 2, [1, 10, 150, 2500, 43750, 787500, 14437500]]
[24, 4, [1, 6, 90, 1620, 31590, 644436, 13533156]]
[25, 5, [1, 5, 75, 1375, 27500, 577500, 12512500]]
[27, 9, [1, 3, 45, 855, 17955, 398601, 9167823]]
[28, 2, [1, 14, 294, 6860, 168070, 4235364, 108707676]]
[32, 16, [1, 2, 34, 748, 18326, 476476, 12864852]]
[36, 6, [1, 6, 126, 3276, 93366, 2800980, 86830380]]
[40, 4, [1, 10, 250, 7500, 243750, 8287500, 290062500]]
[44, 2, [1, 22, 726, 26620, 1024870, 40584852, 1636922364]]
[45, 3, [1, 15, 450, 15750, 590625, 23034375, 921375000]]
[48, 8, [1, 6, 162, 5508, 206550, 8179380, 335354580]]
[49, 7, [1, 7, 196, 6860, 264110, 10722866, 450360372]]
[50, 5, [1, 10, 300, 11000, 440000, 18480000, 800800000]]
[52, 2, [1, 26, 1014, 43940, 1999270, 93565836, 4459971516]]
[54, 9, [1, 6, 180, 6840, 287280, 12755232, 586740672]]
[56, 4, [1, 14, 490, 20580, 936390, 44572164, 2184036036]]
[60, 2, [1, 30, 1350, 67500, 3543750, 191362500, 10524937500]]
[63, 3, [1, 21, 882, 43218, 2268945, 123884397, 6937526232]]
[64, 32, [1, 2, 66, 2860, 138710, 7157436, 384115732]]
[68, 2, [1, 34, 1734, 98260, 5846470, 357803964, 22303113756]]
[72, 12, [1, 6, 234, 11700, 649350, 38181780, 2329088580]]
[75, 5, [1, 15, 675, 37125, 2227500, 140332500, 9121612500]]
[76, 2, [1, 38, 2166, 137180, 9122470, 623976948, 43470394044]]
[80, 8, [1, 10, 450, 25500, 1593750, 105187500, 7187812500]]
[81, 27, [1, 3, 126, 6930, 426195, 27873153, 1895374404]]
[84, 2, [1, 42, 2646, 185220, 13613670, 1029193452, 79247895804]]
[88, 4, [1, 22, 1210, 79860, 5709990, 427107252, 32887258404]]
[90, 3, [1, 30, 1800, 126000, 9450000, 737100000, 58968000000]]
[92, 2, [1, 46, 3174, 243340, 19588870, 1621958436, 136785161436]]
[96, 16, [1, 6, 306, 20196, 1484406, 115783668, 9378477108]]
[98, 7, [1, 14, 784, 54880, 4225760, 343131712, 28823063808]]
[99, 3, [1, 33, 2178, 167706, 13835745, 1187106921, 104465409048]]
[100, 10, [1, 10, 550, 38500, 2983750, 244667500, 20796737500]]
[104, 4, [1, 26, 1690, 131820, 11138790, 984669036, 89604882276]]
[108, 18, [1, 6, 342, 25308, 2087910, 182900916, 16643983356]]
[112, 8, [1, 14, 882, 69972, 6122550, 565723620, 54120892980]]
[116, 2, [1, 58, 5046, 487780, 49509670, 5168809548, 549616748604]]
[117, 3, [1, 39, 3042, 276822, 26990145, 2736800703, 284627273112]]
[120, 4, [1, 30, 2250, 202500, 19743750, 2013862500, 211455562500]]
[121, 11, [1, 11, 726, 61226, 5724631, 566738469, 58185149484]]
[124, 2, [1, 62, 5766, 595820, 64646470, 7214546052, 820053401244]]
[125, 25, [1, 5, 325, 27625, 2624375, 265061875, 27831496875]]
[126, 3, [1, 42, 3528, 345744, 36303120, 3964300704, 444001678848]]
[128, 64, [1, 2, 130, 11180, 1078870, 110907836, 11867138452]]
[132, 2, [1, 66, 6534, 718740, 83014470, 9862119036, 1193316403356]]
[135, 9, [1, 15, 1125, 106875, 11221875, 1245628125, 143247234375]]
[136, 4, [1, 34, 2890, 294780, 32573190, 3765460764, 448089830916]]
[140, 2, [1, 70, 7350, 857500, 105043750, 13235512500, 1698557437500]]
[144, 24, [1, 6, 450, 44100, 4828950, 562089780, 68012863380]]
[147, 7, [1, 21, 1764, 185220, 21392910, 2605656438, 328312711188]]
[148, 2, [1, 74, 8214, 1013060, 131191270, 17474677164, 2370731201916]]
[150, 5, [1, 30, 2700, 297000, 35640000, 4490640000, 583783200000]]
[152, 4, [1, 38, 3610, 411540, 50825190, 6566614548, 873359734884]]
[153, 3, [1, 51, 5202, 619038, 78927345, 10465765947, 1423344168792]]
[156, 2, [1, 78, 9126, 1186380, 161940870, 22736498148, 3251319235164]]
[160, 16, [1, 10, 850, 93500, 11453750, 1488987500, 201013312500]]
[162, 27, [1, 6, 504, 55440, 6819120, 891940896, 121303961856]]
[164, 2, [1, 82, 10086, 1378420, 197803270, 29195762652, 4389096318684]]
[168, 4, [1, 42, 4410, 555660, 75847590, 10831035852, 1592162270244]]
[169, 13, [1, 13, 1183, 138411, 17993430, 2479494654, 354567735522]]
[171, 3, [1, 57, 6498, 864234, 123153345, 18251325729, 2774201510808]]
[172, 2, [1, 86, 11094, 1590140, 239316070, 37046127636, 5840939457276]]
[175, 5, [1, 35, 3675, 471625, 66027500, 9706042500, 1472083112500]]
[176, 8, [1, 22, 2178, 271524, 37334550, 5420976660, 814953491220]]
[180, 6, [1, 30, 3150, 409500, 58353750, 8753062500, 1356724687500]]
[184, 4, [1, 46, 5290, 730020, 109137990, 17069181636, 2748138243396]]
[188, 2, [1, 94, 13254, 2076460, 341577670, 57794941764, 9959994963996]]
[189, 9, [1, 21, 2205, 293265, 43109955, 6699287007, 1078585208127]]
[192, 32, [1, 6, 594, 77220, 11235510, 1739256948, 280020368628]]
[196, 14, [1, 14, 1470, 198940, 29940470, 4778499012, 791638002988]]
[198, 3, [1, 66, 8712, 1341648, 221371920, 37987421472, 6685786179072]]
[200, 20, [1, 10, 1050, 143500, 21883750, 3545167500, 596769862500]]
[204, 2, [1, 102, 15606, 2653020, 473564070, 86946363252, 16258969928124]]
[207, 3, [1, 69, 9522, 1533042, 264449745, 47442284253, 8729380302552]]
[208, 8, [1, 26, 3042, 448188, 72830550, 12497722380, 2220428676180]]
[212, 2, [1, 106, 16854, 2977540, 552333670, 105385264236, 20479869683196]]
[216, 36, [1, 6, 666, 97236, 15898086, 2766266964, 500694320484]]
[220, 2, [1, 110, 18150, 3327500, 640543750, 126827662500, 25576911937500]]
[224, 16, [1, 14, 1666, 256564, 44000726, 8008132132, 1513536972948]]
[225, 15, [1, 15, 1800, 279000, 48127500, 8807332500, 1673393175000]]
[228, 2, [1, 114, 19494, 3703860, 738920070, 151626398364, 31689917258076]]
[232, 4, [1, 58, 8410, 1463340, 275839590, 54395567148, 11042300131044]]
[234, 3, [1, 78, 12168, 2214576, 431842320, 87577622496, 18216145479168]]
[236, 2, [1, 118, 20886, 4107580, 848215270, 180160923348, 38974813084284]]
[240, 8, [1, 30, 4050, 688500, 129093750, 25560562500, 5239915312500]]
[242, 11, [1, 22, 2904, 489808, 91594096, 18135631008, 3723849566976]]
[243, 81, [1, 3, 369, 60147, 11006901, 2146345695, 435708176085]]
[244, 2, [1, 122, 22326, 4539620, 969208870, 212838267852, 47604825909564]]
[245, 7, [1, 35, 4900, 857500, 165068750, 33508956250, 7036880812500]]
[248, 4, [1, 62, 9610, 1787460, 360173190, 75924508452, 16475618334084]]
[250, 25, [1, 10, 1300, 221000, 41990000, 8481980000, 1781215800000]]
[252, 6, [1, 42, 6174, 1123668, 224171766, 47076070860, 10215507376620]]
[256, 128, [1, 2, 258, 44204, 8509270, 1746102204, 373083837588]]
[260, 2, [1, 130, 25350, 5492500, 1249543750, 292393237500, 69687054937500]]
[261, 3, [1, 87, 15138, 3073014, 668380545, 151187679279, 35075541592728]]
[264, 4, [1, 66, 10890, 2156220, 462509190, 103787062236, 23974811376516]]
[268, 2, [1, 134, 26934, 6015260, 1410578470, 340231526964, 83583545124156]]
[270, 9, [1, 30, 4500, 855000, 179550000, 39860100000, 9167823000000]]
[272, 8, [1, 34, 5202, 1002252, 212978550, 47792386620, 11103764491380]]
[275, 5, [1, 55, 9075, 1830125, 402627500, 93006952500, 22166657012500]]
[276, 2, [1, 138, 28566, 6570180, 1586698470, 394135899948, 99716382686844]]
[279, 3, [1, 93, 17298, 3753666, 872727345, 211025472021, 52334317061208]]
[280, 4, [1, 70, 12250, 2572500, 585243750, 139288012500, 34125563062500]]
[284, 2, [1, 142, 30246, 7158220, 1778817670, 454665796452, 118364662343004]]
[288, 48, [1, 6, 882, 171108, 37215990, 8619223284, 2077232811444]]
[289, 17, [1, 17, 2601, 515865, 114006165, 26745846309, 6517071217293]]
[292, 2, [1, 146, 31974, 7780340, 1987876870, 522414041436, 139832825091036]]
[294, 7, [1, 42, 7056, 1481760, 342286560, 83381006016, 21012013516032]]
[296, 4, [1, 74, 13690, 3039180, 730922790, 183900173964, 47630145056676]]
[297, 9, [1, 33, 5445, 1138005, 262879155, 64195089651, 16241357681703]]
[300, 10, [1, 30, 4950, 1039500, 241683750, 59454202500, 15160821637500]]
[304, 8, [1, 38, 6498, 1399236, 332318550, 83345492340, 21642046177620]]
[306, 3, [1, 102, 20808, 4952304, 1262837520, 334904510304, 91094026802688]]
[308, 2, [1, 154, 35574, 9130660, 2460712870, 682109607564, 192582279202236]]
[312, 4, [1, 78, 15210, 3559140, 902241990, 239274575748, 65321959179204]]
[315, 3, [1, 105, 22050, 5402250, 1418090625, 387138740625, 108398847375000]]
[316, 2, [1, 158, 37446, 9860780, 2726505670, 775418212548, 224612808901404]]
[320, 32, [1, 10, 1650, 357500, 86693750, 22366987500, 6001808312500]]
[324, 54, [1, 6, 990, 215820, 52767990, 13740784596, 3723752625516]]
[325, 5, [1, 65, 12675, 3020875, 785427500, 214421707500, 60395447612500]]
[328, 4, [1, 82, 16810, 4135260, 1102046790, 307250645052, 88180935129924]]
[332, 2, [1, 166, 41334, 11435740, 3322082470, 992638242036, 302092904992956]]
[333, 3, [1, 111, 24642, 6382278, 1771082145, 511134307047, 151295754885912]]
[336, 8, [1, 42, 7938, 1889244, 495926550, 137470839660, 39454130982420]]
[338, 13, [1, 26, 4732, 1107288, 287894880, 79343828928, 22692335073408]]
[340, 2, [1, 170, 43350, 12282500, 3654043750, 1118137387500, 348486152437500]]
[342, 3, [1, 114, 25992, 6913872, 1970453520, 584042423328, 177548896691712]]
[343, 49, [1, 7, 1225, 282975, 73290525, 20213526795, 5801282190165]]
[344, 4, [1, 86, 18490, 4770420, 1333332390, 389866390836, 117349783641636]]
[348, 2, [1, 174, 45414, 13170060, 4010283270, 1256020720164, 400670609732316]]
[350, 5, [1, 70, 14700, 3773000, 1056440000, 310593360000, 94213319200000]]
[351, 9, [1, 39, 7605, 1878435, 512812755, 147997761093, 44251330566807]]
[352, 16, [1, 22, 4114, 995588, 268310966, 76736936276, 22790870073972]]
[356, 2, [1, 178, 47526, 14099380, 4391956870, 1407182981148, 459210712847964]]
[360, 12, [1, 30, 5850, 1462500, 405843750, 119318062500, 36392009062500]]
[361, 19, [1, 19, 3610, 891670, 245655085, 71878677871, 21851118072784]]
[363, 11, [1, 33, 6534, 1653102, 463695111, 137717447967, 42416973973836]]
[364, 2, [1, 182, 49686, 15071420, 4800247270, 1572561005652, 524711188885884]]
[368, 8, [1, 46, 9522, 2482068, 713594550, 216647305380, 68099469657780]]
[369, 3, [1, 123, 30258, 8684046, 2670344145, 853976057571, 280104146883288]]
[372, 2, [1, 186, 51894, 16087140, 5236364070, 1753134690636, 597818929506876]]
[375, 25, [1, 15, 2925, 745875, 212574375, 64410035625, 20289161221875]]
[376, 4, [1, 94, 22090, 6229380, 1903075590, 608222958564, 200105353367556]]
[378, 9, [1, 42, 8820, 2346120, 689759280, 214377184224, 69029453320128]]
[380, 2, [1, 190, 54150, 17147500, 5701543750, 1949927962500, 679224906937500]]
[384, 64, [1, 6, 1170, 301860, 87388470, 26950604148, 8651143931508]]
[387, 3, [1, 129, 33282, 10017882, 3230766945, 1083599233353, 372758136273432]]
[388, 2, [1, 194, 56454, 18253460, 6197049670, 2164009744764, 769666132554396]]
[392, 28, [1, 14, 2842, 755972, 224901670, 71158888388, 23411274279652]]
[396, 6, [1, 66, 15246, 4360356, 1366971606, 451100629980, 153825314823180]]
[400, 40, [1, 10, 2050, 553500, 167433750, 53913667500, 18061078612500]]
[404, 2, [1, 202, 61206, 20606020, 7284228070, 2648545326252, 980844619155324]]
[405, 27, [1, 15, 3150, 866250, 266371875, 87103603125, 29615225062500]]
[408, 4, [1, 102, 26010, 7959060, 2638428390, 915006965652, 326657486737764]]
[412, 2, [1, 206, 63654, 21854540, 7878561670, 2921370667236, 1103304321992796]]
[414, 3, [1, 138, 38088, 12264336, 4231195920, 1518153096096, 558680339363328]]
[416, 16, [1, 26, 5746, 1643356, 523408886, 176912203468, 62096183417268]]
[420, 2, [1, 210, 66150, 23152500, 8508543750, 3216229537500, 1238248371937500]]
[423, 3, [1, 141, 39762, 13081698, 4611298545, 1690502046597, 635628769520472]]
[424, 4, [1, 106, 28090, 8932620, 3077287590, 1109054447436, 411459199998756]]
[425, 5, [1, 85, 21675, 6755375, 2296827500, 819967417500, 302021332112500]]
[428, 2, [1, 214, 68694, 24500860, 9175572070, 3534430361364, 1386674845108476]]
[432, 72, [1, 6, 1314, 381060, 124035030, 43015348404, 15528540773844]]
[436, 2, [1, 218, 71286, 25900580, 9881071270, 3877332366348, 1549640502417084]]
[440, 4, [1, 110, 30250, 9982500, 3568743750, 1334710162500, 513863412562500]]
[441, 21, [1, 21, 4851, 1460151, 490610736, 175148032752, 64979920150992]]
[444, 2, [1, 222, 73926, 27352620, 10626492870, 4246346550852, 1728263046196764]]
[448, 32, [1, 14, 3234, 980980, 333042710, 120295026852, 45190831754068]]
[450, 15, [1, 30, 7200, 2232000, 770040000, 281834640000, 107097163200000]]
[452, 2, [1, 226, 76614, 28857940, 11413315270, 4642936651836, 1923723419410716]]
[456, 4, [1, 114, 32490, 11111580, 4116840390, 1595687335164, 636679246730436]]
[459, 9, [1, 51, 13005, 4200615, 1499619555, 565956420057, 221288960242287]]
[460, 2, [1, 230, 79350, 30417500, 12243043750, 5068620112500, 2137268147437500]]
[464, 8, [1, 58, 15138, 4975356, 1803566550, 690405275340, 273630624126420]]
[468, 6, [1, 78, 21294, 7197372, 2666626326, 1039984267140, 419113659657420]]
[472, 4, [1, 118, 34810, 12322740, 4725770790, 1895979240948, 783039426511524]]
[475, 5, [1, 95, 27075, 9431125, 3583827500, 1429947172500, 588661586012500]]
[476, 2, [1, 238, 84966, 33703180, 14037374470, 6013611222948, 2623939030279644]]
[477, 3, [1, 159, 50562, 18758502, 7456504545, 3082518978903, 1306988047054872]]
[480, 16, [1, 30, 7650, 2524500, 927753750, 361823962500, 146538704812500]]
[484, 22, [1, 22, 5566, 1836780, 676853430, 265055803188, 107877711897516]]
[486, 81, [1, 6, 1476, 481176, 176110416, 68683062240, 27885323269440]]
[488, 4, [1, 122, 37210, 13618860, 5399877990, 2239869390252, 956424229637604]]
[490, 7, [1, 70, 19600, 6860000, 2641100000, 1072286600000, 450360372000000]]
[492, 2, [1, 246, 90774, 37217340, 16022064870, 7094570324436, 3199651216320636]]
[495, 3, [1, 165, 54450, 20963250, 8647340625, 3709709128125, 1632272016375000]]
[496, 8, [1, 62, 17298, 6077364, 2354978550, 963657222660, 408269443333620]]
[500, 50, [1, 10, 2550, 858500, 324083750, 130281667500, 54501164237500]]
[504, 12, [1, 42, 11466, 4013100, 1559089350, 641721176460, 274014942348420]]
[507, 13, [1, 39, 10647, 3737097, 1457467830, 602517200922, 258479879195538]]
[508, 2, [1, 254, 96774, 40967660, 18210124870, 8325669090564, 3876986573172636]]
[512, 256, [1, 2, 514, 175788, 67590486, 27712099260, 11833066384020]]
[513, 9, [1, 57, 16245, 5864445, 2339913555, 986975537499, 431308309887063]]
[516, 2, [1, 258, 99846, 42933780, 19384601670, 9002209015548, 4258044864354204]]
[520, 4, [1, 130, 42250, 16477500, 6961743750, 3077090737500, 1400076285562500]]
[522, 3, [1, 174, 60552, 24584112, 10694088720, 4838005736928, 2244834661934592]]
[524, 2, [1, 262, 102966, 44961820, 20614994470, 9722031392052, 4669815745315644]]
[525, 5, [1, 105, 33075, 12733875, 5348227500, 2358568327500, 1073148589012500]]
[528, 8, [1, 66, 19602, 7331148, 3024098550, 1317297328380, 594101095099380]]
[529, 23, [1, 23, 6348, 2287396, 920676890, 393865573542, 175138891701676]]
[531, 3, [1, 177, 62658, 25877754, 11450906145, 5269707007929, 2487301707742488]]
[532, 2, [1, 266, 106134, 47052740, 21903050470, 10487180565036, 5114248388882556]]
[536, 4, [1, 134, 44890, 18045780, 7858937190, 3580531783764, 1679269406585316]]
[539, 7, [1, 77, 23716, 9130660, 3866834510, 1726928292166, 797840870980692]]
[540, 18, [1, 30, 8550, 3163500, 1304943750, 571565362500, 260062239937500]]
[544, 16, [1, 34, 9826, 3674924, 1530605846, 676527783932, 310526252824788]]
[548, 2, [1, 274, 112614, 51427060, 24659275270, 12161954563164, 6109355175562716]]
[549, 3, [1, 183, 66978, 28599606, 13084319745, 6225519334671, 3038053435319448]]
[550, 5, [1, 110, 36300, 14641000, 6442040000, 2976222480000, 1418666048800000]]
[552, 4, [1, 138, 47610, 19710540, 8840177190, 4147811137548, 2003392779435684]]
[556, 2, [1, 278, 115926, 53712380, 26131072870, 13075988864148, 6664395657760764]]
[558, 3, [1, 186, 69192, 30029328, 13963637520, 6752815104672, 3349396291917312]]
[560, 8, [1, 70, 22050, 8746500, 3826593750, 1767886312500, 845638952812500]]
[564, 2, [1, 282, 119286, 56064420, 27667791270, 14044170848652, 7260836328753084]]
[567, 27, [1, 21, 6174, 2376990, 1023294195, 468464082471, 222988903256196]]
[568, 4, [1, 142, 50410, 21474660, 9910555590, 4784816238852, 2378053670709444]]
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